Azam Zaka
1
and Ahmad Saeed Akhtar
2
1
Govt. College Ravi Road Shahdara, Lahore. (azamzka@gmail.com)
2
College of Statistical & Actuarial Sciences, University of the Punjab, Lahore
Abstract
This paper is concerned with estimation of the scale and shape parameters of the Power function
distribution through the use of symmetrically located percentiles from a sample. The process requires
algebraic solution of two equations derived from the cumulative distribution function. Sampling behavior
of the estimates is indicated by a Monte Carlo simulation. Three alternative methods of estimated
distribution function are compared, for precision and variability, with maximum likelihood (M.L.E), Ridge
Regression and Least squares estimators (L.S.E). The best percentile estimator (using the 5
th
and 95
th
) is
inferior to M.L.E in variability. The said estimator is also substandard to least square estimator in accuracy
and variability to a small degree.
Keywords: Percentiles, least squares method, ridge regression, Monte Carlo study, total deviation, mean square
error.
I. INTRODUCTION
The Power function distribution is a flexible life time distribution model that may offer a good fit to some sets of
failure data. Theoretically, the Power function distribution is a special case of Pareto distribution. An excellent
account of this distribution and its properties is given in Merran et al. [1]. Meniconi and Barry [2] discussed the
application of Power function distribution. They proved that the Power function distribution is the best distribution to
check the reliability of any electrical component. They used Exponential distribution, Lognornal distribution and
Weibull distribution and showed from reliability and hazard function that Power function distribution is the best
distribution.
The probability distribution of Power function distribution is
f ( t ) =
γt
γ −1
β
γ
; 0 <t < β (1)
With shape parameter γ and scale parameter β , the interval (0, β )
Moments of order statistics for a Power function distribution were calculated by Malik [3]. Lwin [4] discussed
Bayesian estimation for the scale parameter of the pareto distribution using a Power function Prior. Ahsanullah [5]
discussed the estimation of the location and scale parameters of a Power function distribution by linear function of
the order statistics. Ahsanullah [6] considered the estimation of the parameters of a Power function distribution by
record values. Saran [7] used kth record values for the estimation of parameters of a Power function distribution and
its characterization. Tavangar [8] characterized the estimation of the Power function distribution by dual generalized
order statistics. Munawar and Farooq [9] proposed the Bayesian parameters estimation of hybrid censored power
function distribution under different loss functions.
Cohen and Whitten [10] used the moment and modified moment estimators for the Weibull distribution. Samia and
Mohamed [11] used five modifications of moments to estimate the parameters of the Pareto distribution. Lalitha and
Anand [12] used modified maximum likelihood to estimate the scale parameter of the Rayleigh distribution. Rafiq et
[1]
Estimation of Parameters of the Power Function Distribution: Different
Classical Methods
al. [13] discussed the parameters of the Gamma distribution. Rafiq [14] discussed the method of fractional moments
to estimate the parameters of Weibull distribution. Kang and Young [15] estimated the parameters of a Pareto
distribution by jackknife and bootstrap methods. Neil [16] utilized the common percentiles for estimation of
parameters of the Weibull distribution. Dereny and Rashwan [17] addressed the problem of solving multicollinearity
problems using different ridge regression models.
In this paper, we use the Percentiles method, least squares and ridge regression to estimate the two parameter of the
Power function distribution. The paper introduces the ridge regression estimators by taking the value of “K = 0.1”.
Also, three alternatives of estimated distribution function are compared for precision and variability with maximum
likelihood (M.L.E), least squares estimators (L.S.E) and percentile estimators (P.E). We examined the performance
these methods using two parameters Power function distribution to find the most accurate method (the method
which has least M.S.E).
II. METHODOLOGY
A. Percentile Estimator (P.E)
Let t
1
,t
2
,t
3
,…,t
n
be a random sample of size n drawn from probability density function of Power function
distribution. The cumulative distribution function for a Power function distribution with shape and scale parameters
β ∧ γ, respectively
F
(
t
i
)
=
(
t
i
β
)
γ
( 2 )
By solving we get
t
i
= β ( R
i
)
1
γ
( 3 )
Where R
i
= F
(
t
i
)
Let P75 and P25 are used. therefore ( 3 ) becomes
P
75
= β ( 0.75 )
1
γ
( 4 )
P
25
= β ( 0.25 )
1
γ
( 5 )
Dividing (4) and (5) we get
(
P
75
P
25
)
γ
=
(
0.75
0.25
)
γ ln
(
P
75
P
25
)
=ln
(
0.75
0.25
)
^ γ =
ln
(
0.75
0.25
)
ln
(
P
75
P
25
)
∧
^
β =
P
75
( .75 )
1
^ γ
generally ^ γ =
ln
(
H
L
)
ln
(
P
H
P
L
)
( 6 )
[2]
Estimation of Parameters of the Power Function Distribution: Different Classical Methods
Azam Zaka1 and Ahmad Saeed Akhtar2
1 Govt. College Ravi Road Shahdara, Lahore. (azamzka@gmail.com)
2College of Statistical & Actuarial Sciences, University of the Punjab, Lahore
Abstract
This paper is concerned with estimation of the scale and shape parameters of the Power function distribution through the use of symmetrically located percentiles from a sample. The process requires algebraic solution of two equations derived from the cumulative distribution function. Sampling behavior of the estimates is indicated by a Monte Carlo simulation. Three alternative methods of estimated distribution function are compared, for precision and variability, with maximum likelihood (M.L.E), Ridge Regression and Least squares estimators (L.S.E). The best percentile estimator (using the 5th and 95th) is inferior to M.L.E in variability. The said estimator is also substandard to least square estimator in accuracy and variability to a small degree.
Keywords: Percentiles, least squares method, ridge regression, Monte Carlo study, total deviation, mean square error.
INTRODUCTION
The Power function distribution is a flexible life time distribution model that may offer a good fit to some sets of failure data. Theoretically, the Power function distribution is a special case of Pareto distribution. An excellent account of this distribution and its properties is given in Merran et al. [1]. Meniconi and Barry [2] discussed the application of Power function distribution. They proved that the Power function distribution is the best distribution to check the reliability of any electrical component. They used Exponential distribution, Lognornal distribution and Weibull distribution and showed from reliability and hazard function that Power function distribution is the best distribution.
The probability distribution of Power function distribution is
[1]
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